According to the theory of relativity, time dilation occurs as an object's speed approaches the speed of light. As an object accelerates to relativistic speeds, time appears to slow down from the perspective of a stationary observer. This phenomenon has been experimentally verified and is a fundamental aspect of our understanding of the universe.
However, it's important to note that time dilation affects the perception of time for the moving object, not for the observer in the same reference frame as the object. So if you were traveling at a high speed, time would appear to pass normally for you. It is only from the perspective of an outside observer that your time would seem to be dilated.
As for the possibility of traveling at speeds close to or exceeding the speed of light, according to our current understanding of physics, it becomes increasingly difficult as you approach the speed of light. As you approach that limit, the energy required to accelerate further also increases significantly. This is due to the concept of relativistic mass increase, which means that an object's mass appears to increase as it approaches the speed of light.
Thus, while it is theoretically possible to approach the speed of light, it becomes increasingly challenging to accelerate further as you get closer to that limit. Nevertheless, it's important to note that our understanding of the laws of physics is still evolving, and there may be future breakthroughs or new discoveries that could provide insights into faster-than-light travel.